(m) 1 11 (m 1) (s) (s)
0 0
y =f (t, y, y , y , ..., y − )y (t )=y , s=0(1)m 1−
1 n
0
y′ =f (y), y(t )= µ ∈, f C [a, b], y, t∈R
1 n
y′′=f (x, y, y ), y(a)′ = µ, y (a)′ = τ ∈, f C [a, b], y, x∈R
k 1 k
k j n j j n j
j 0 j 2
n u
y (t) (t)y (t)f
(t)f t (0, 1] and v (1, 2) −
+ +
= =
+
= α + β
+τ ∈ ∈
∑
∑
2k j j j 0
m j
y(x) a x
a , j 0(1)2k, y C (a, b) P(x) =
=
∈ℜ = ∈ ⊂
∑
2k j 1 j j 1 y (x) ja x−
= ′ =
∑
2k
j 2 j j 2
y (x) j( j 1)a x− =
′′ =
∑
−© Medwell Journals, 2007
A Zero Stable Continuous Hybrid Methods for Direct Solution
of Second Order Differential Equations
S.J. Kayode
Department of Mathematical Sciences, Federal University of Technology, Akure
Abstract: This study produces a zero stable hybrid three-step methods for a direct solution of general second order ordinary differential Eq of form y” = f (x, y y’). The differential system from the basis polynomial function to the problem is collocated at all the grid points and at an off-grid point. The basis function is interpolated at x , I = 0,1,2. The method is consistent and zero-stable. The efficiency and accuracy of the method are shownn+i
with some test examples.
Key words: Collocation, differential system, basis function, hybrid, symmetric, continuous method, zero stable
INTRODUCTION MATERIALS AND METHODS
The solution of higher order differential Eq of the form
(1)
is considered in this study. It has been observed in literature that solutions of such Eq are usually reduced to system of first order Eq of the form
(2)
There are numerous numerical methods developed to handle the reduced Eq. 2 (Lambert, 1973; Goult et al. 1973; Jain, 1984; Lxaru, 1984; Jacques and Judd, 1987; Fatunla, 1988; Bun and Vasil’ Yer, 1992; Awoyemi, 1992; Jaun, 2001; Chan et al., 2004). This approach has many disadvantages such as much of computational burden and computer time wastage. Hence, there is need for direct methods for solving Eq. 1 without reducing it to system of first order equations.
Awoyemi and Kayode (2003) highlighted some of the direct methods for solving (1), in which m = 2 and the derivative is absent in the right side.
In this study, a three-step hybrid numerical technique is proposed for a direct solution of initial value problems (1) in which m=2 to be of the form
(3)
In this study, the development of the collocation methods for the solution of second order ordinary differential Eq. 3 directly without reducing it to first order system of Eq. is discussed. The method obtained is an order five hybrid linear multistep with continuous coefficients of the form
(4)
The approximate solution to problem (1) is taken to be a partial sum of a P- series of a single variable x in the form
(5)
It is assumed that the initial value problem (1) satisfies the hypotheses of existence and uniqueness theorem. The first and second derivatives of (5) are respectively taken as
(6)
2k
j 2 j j 2
j( j 1)a x− f (x, y(x), y (x)) = ′ − =
∑
2k j 2 j n i n i j 2j( j 1)a x−+ f +, i 0(1)k =
− = =
∑
2k
j 2 j n i n v j 2
j( j 1)a x−+ f + , v (1, 2) =
− = ∈
∑
2k j j n i n i j 0
a x + y +, i 0(1)k 1 =
= = −
∑
n i n i n i n i n i
n i n i f f (x , y , y )y is the numerical
approximation to y(x ) at x
+ + + + +
+ +
′ =
n i n
x + =x +ih.
k 1 k
k j n j j n j n u
j 0 j 2
y (x) (x)y (x)f (x)f
−
+ + +
= =
=
∑
α +∑
β + τn k 1
1 dt 1
t (x x ) and t (0, 1]
h + − dx h
= − = ∈
2(t) {1 t}
α = +
1(t) t
α = −
0(t) 0
α =
2
3 3
4 5 6
h
{(8v 13)t 20(2 v)t 360(3 v)
5(8 3v)t 3(5 v)t 2t }
β = − + −
−
+ − + − +
2
2 2
3 4 5 6
h
{(75 43v)t 60(2 v)t 120(2 v)
10(4 v)t 5(2v 3)t 3(v 4)t 2t }
β = − + − +
−
− + − + − −
2 v
3 4 5 6
h
{11t 60v(3 v)(2 v)(v 1)
20t 5t 6t 2t }
β =
− − −
− − + +
2 1
3 4 5 6
h
{11(2v 3)t 20(2 120(v 1)
v)t 5vt 3(v 3)t 2t }
β = − +
−
− + + − −
2
3 .
4 5 6
h
{(11 7v)t 10(v 2)t 360v
5t 3(2 v)t 2t }
β = − + −
− + − + 2 1 1 h 1 h ′ α = ′ α = −
2 3
3 4 5
h
{(8v 13) 60(2 v)t 360(3 v)
20(4 3v)t 15(5 v)t 12t
′
β = − + − +
−
− + − +
2
2 3 4 5
h
{(75 43v) 120(2 v)t 120(2 v)
30(4 v)t 20(2v 3)t 15(v 4)t 12t }
′
β = − + − +
−
− + − + − −
2 3 4 5
v
h
{11 60t 20t 30t 12t }
60v(3 v)(2 v)(v 1)
′
β = − − + +
− − −
2 1
3 4 5
h
{11(2v 3) 60(2 v)t 120(v 1)
20vt 15(v 3)t 12t }
′
β = − + −
−
+ + − −
2 3 4 5
0 h
{(11 7v) 30(v 2)t 40t 15(2 v)t 12t } 360v
′
β = − + − − + − +
2 n 3 n 2 n 1
n 3 n 2 n v n 1 n
h
y 2 y
60v(3 v)(2 v)(v 1)
(Af Bf Cf Df Ef )
+ + +
+ + + +
− + =
− − −
+ + + +
From (3) and (7), we have
(8)
Thus, collocating Eq. 8 at the grid points x , I = 0,1,2,3,v,n+i
1<v<2, and interpolating (5) at x , I = 0(1)k-1, for k = 3,n+i
yields a system of Eq.
(9)
(10)
(11)
whe re
and
Solving Eq. 9, 10 and 11 to obtain the parameters a ‘s, jj , and then substituting for these values into Eq. 3 produces a continuous method expressed as
(12)
Using the transformations
(13)
the coefficients in the continuous method (12) are obtained, as a function of t, to be
(14)
Taking the first derivatives of aj, $j, in (14) yields
(15)
To obtain a sample discrete scheme from the continuous method (12), the values of t in (14) could be taken in the interval I = (0, 1]. Hence for the purpose of this research t is taken to be 1, which implies that x = xn+3 from (13), to have a one-point hybrid discrete scheme as
A v(14 5v)(2 v)(v 1) B v(103 50v)(3 v)(v 1)
C 6
D v(5v 2)(3 v)(2 v) E (3 v)(2 v)(v 1)
= − − −
= − − −
= −
= − − −
= − − − −
n 3 n 2 n 1
n 3 n 2 n v n 1
1 h
y (y y )
h 360v(3 v)(2 v)(v 1) (Ff Gf Hf If Jf }
+ + +
+ + + +
′ = − +
− − −
+ + + +
F v(354 127v)(2 v)(v 1) G 3v(303 138v)(3 v)(v 1)
H 162 I 9v(10 v)(3 v)(2 v) J (8v 27)(3 v)(2 v)(v 1)
= − − − = − − − = − = − − − = − − − − 2
n 3 n 2 n 1 n 3 n 2
5 n 1 n
n 4
h
y 2 y (155f 1890f
2100
512f 595f 284f )
+ + + + + + + = − + + − + − p 2
of order P = 5, error constant C + ≈ −0.002292
n 3 n 2 n 1 n 3
n 2 5 n 1 n
n 4
1 h
y (y y ) (3905f
h 12600
18270f 13824f 11025f 476f )
+ + + +
+ + +
′ = − + +
− + −
p 2
Order P = 5,C+ =0.006344
2
n 3 n 2 n 1 n 3
n 2 3 n 1 n
n 2
h
y 2 y (13f
180
168f 32f 33f 2f )
+ + + +
+ + +
= − + +
− + −
p 2
Order P = 5, Error constant C+ ≈ −0.002083
n 3 n 2 n 1 n 3
n 2 3 n 1 n
n 2
1 h
y (y y ) (109f
h 360
576f 288f 153f 10f )
+ + + +
+ + +
′ = − + +
− + −
p 2
Order P = 5, Error constant C+ ≈0.005407
2
n 3 n 2 n 1 n 3 n 2
7 n 1 n
n 4
h
y 2 y (147f 2170f
2100
512f 315f 20f )
+ + + + + + + = − + + − + − p 2
Order P = 5, Error constant C + ≈ −0.001875
2
n 3 n 2 n 1 n 3 n 2
7 n 1 n
n 4
1 h
y ' (y y ) (3689f 25830f
h 12600
13824f 3465f 260f )
+ + + + + + + = − + + − + − p 2
Order P = 5, Error constant C+ ≈0.004469
2
n 3 n 2 n 1 n n 2
5 n 1 n
n 4
23 29 31 h
y y y y (495f
4 2 4 240
512f 1990f 127f )
+ + + + + + = − + − + − + + p 2
having order p = 5 and C+ ≈0.0122396 and
n 3 n 2 n 1 n
n 2 5 n 1 n
n 4
1 h
y { 757y 1538y 781y }
24h 7200
{45585f 65024f 248410f 16129f }
+ + +
+ + +
′ = − + − +
− + +
p 2
p = 5, C + ≈ −0.054671
2
n 3 n 2 n 1 n
n 2 3 n 1 n
n 2
9 13 h
y y 12y y
2 2 24
(51f 32f 150f 11f )
+ + +
+ + +
= − + − +
− + +
p 2
of order p = 5 and C + ≈0.011458
n 3 n 2 n 1 n
n 2 3 n 1 n
n 2
1 h
y { 105y 214y 109y }
4h 720
{4749f 4064f 18618f 1397f }
+ + +
+ + +
′ = − + − +
− + +
p 2
and of order p = 5,C + ≈ −0.051364
where for v = 7/4:
and from (15)
(17)
where
Taking the values of v in (16) and (17) are taken at three points 5/4, 3/2, 7/4, in the interval (Awoyemi, 1992; Awoyemi and Kayode, 2002) to obtain the following discrete schemes:
For v = 5/4:
(18)
(19)
For: v = 3/2
(20)
(21)
(22)
(23)
Starting values for the methods: The set of implicit
discrete schemes (18), (20) and (22) and their respective first derivatives (19) (21) and (23) are not self-starting. Thus to be able to implement them, some starting values, of the same order p = 5 and their derivatives are developed using the same technique for the main method (13). Thus at t = 1 and r = 5/4, 3/2, 7/4, the main starting values are:
For v = 5/4:
(24)
(25)
For v = 3/2:
(26)
2
n 3 n 2 n 1 n
n 2 7 n 1 n
n 4
13 19 21 h
y y y y
4 2 4 336
(847f 512f 1638f 127f )
+ + +
+ + +
= − + − +
− + +
p 2
of order p = 5 and C+ ≈0.01015625 and
n 3 n 2 n 1 n
n 2 7 n 1 n
n 4
1 h
y { 503y 1030y 527y }
24h 100080
{83377f 65024f 201978f 16129f }
+ + +
+ + +
′ = − + − +
− + +
p 2
also of order p = 5 and C+ ≈ −0.0458519
2
n 2 n 1 n n 1
y + =2y+ −y +h f + , p = 2, cp + 2 = 0.0833
n 2 n 1 n n 1 n
1 h
y (y y ) (11f 2f )
h 6
p = 2, cp + 2 =- 0.375
+ + +
′ = − + −
2 3
n j n n n
4
n n n
n n
n n n
( jh) ( jh)
y y ( jh)y f
2! 3!
f f f
y f O(h )
x y y
+ = + ′ + +
∂ ∂ ∂
+ ′ + +
∂ ∂ ∂ ′
2
n j n n
3
n n n
n n
n n n
( jh) y y ( jh)f
2!
f f f
y f O(h )
x y y
+
′ = ′ + +
∂ ∂ ∂
+ ′ + +
∂ ∂ ∂ ′
5 3 7
where j 1, , and
4 2 4
=
(4x 3) y(x)
32exp( 3x) − =
−
(4x 3)
y (x) y( )
32exp( 3x) 6
1 3 1
y( ) , h
4 6 2 40
− Π
′′ =
− Π
= = =
1 (2 x)
y(x) 1 ln{ }
2 (2 x)
+ = +
−
For7/4: The initial values y , y' are obtained in the given
(28)
(29)
Other starting values foryn+2,yn+2 , yn+v, yn+v, yn+1, yn+1 are obtained to be
(30)
(31)
The initial values yn, yn’ are obtained in the given problem.
(32)
(33)
n n
preblem.
NUMERICAL EXPERIMENT
The accuracy of the continuous method developed for the direct solution of second order ordinary differential Eq tested with the following problems
yO = 2 y , y(1) = 1, y’(1) = -1;3
Theoretical solution: y(x) = 1/x.
yO = y+xe3x
Theoretical solution: .
Theoretical solution is given as y(x) = Sin x.2
YO-x(y') = 0, y(0) = 1, y'(0) = 1/2; h1/402
Theoretical solution is .
RESULTS
[image:4.612.73.541.599.719.2]The absolute errors obtained from the method (15) for k = 3 are compared with those obtained from the method for k = 2 in Kayode (2004) for the problems (i)-(iv). The results are shown in the Table 1 and 2 below. The accuracy of the results is further illustrated graphically in the Fig. 1-4.
Table 1:Comparison of errors for problems (i) and (ii) for k = 2, 3
Table 2:Comparison of errors for problem (iii) and (iv) for k = 2, 3
Kayode (2004) New method (15) Kayode (2004) New method (15) for problem (iii) for problem (iv) for problem (iii) for problem (iv) x Errors for k = 2 Errors for k = 3 x Errors for k = 2 Errors for k = 3 1.1 0.5025381D-05 0.2282106D-05 1.1 0.1053972D-06 0.8047086D-07 1.2 0.6249908D-05 0.2893084D-05 1.2 0.2131542D-06 0.1625604D-06 1.3 0.7288316D-05 0.3453509D-05 1.3 0.3258333D-06 0.2480160D-06 1.4 0.8112927D-05 0.3954212D-05 1.4 0.4463340D-06 0.3387987D-06 1.5 0.8700464D-05 0.4384330D-05 1.5 0.5781210D-06 0.4372248D-06 1.6 0.9032439D-05 0.4731177D-05 1.6 0.7294657D-06 0.5490446D-06 1.7 0.9095639D-05 0.4980477D-05 1.7 0.8987956D-06 0.6725762D-06 1.8 0.8882690D-05 0.5116961D-05 1.8 0.1097337D-05 0.8153498D-06 1.9 0.8392690D-05 0.5125297D-05 1.9 0.1336004D-05 0.9842053D-06 2.0 0.7631870D-05 0.4991312D-05 2.0 0.1630750D-05 0.1188939D-05
Fig 1: Comparison of errors for problem (i) for k = 2, 3 Fig 3: Comparison of errors for problem (iii) for k = 2, 3
Fig 2: Comparison of errors for problem (ii) for k = 2, 3 Fig 4: Comparison of errors for problem (iv) for k = 2, 3
CONCLUSION The comparison of the absolute errors obtained
This study has considered the development of shown in Table 1 and 2 and also in Fig. 1 and 4. These a continuous hybrid numerical method with step errors show a considerable improvement in accuracy of number k = 3. A set of discrete schemes of the the new method over Kayode (2004).
same order p = 5 are obtained from the continuous
method. The major predictors for the methods are REFERENCES
constructed to be of the same order p = 5 with the
methods. The efficiency of the method is Awoyemi, D.O., 1992. On some continuous linear compared with existing order four method (Kayode multistep methods for initial value problems. Ph.D
2004; 2006). Thesis (Unpublished), University of Ilorin, Nigeria.
[image:5.612.296.535.96.488.2] [image:5.612.317.528.416.565.2]Awoyemi, D.O. and S.J. Kayode, 2003. An optimal order Jacques, I. and C.J. Judd, 1987. Numerical Analysis. collocation method for direct solution of initial value
problems of general second order ordinary differential equations. FUTAJEET, 3: 33-40. Bun, R.A. and Y.D. Vasil Yev, 1992. A numerical method
for solving differential equations of any orders. Comp. Math. Phys., 32: 317-330.
Chan, R.P.K., P. Leone and A. Tsai, 2004. Order conditions and symmetry for two-step hybrid methods. Int. J. Comp. Math, 81: 1519-1536. Fatunla, S.O., 1988. Numerical Methods for IVPs in
ordinary differential equations Academic Press Inc. Harcourt Brace Jovanovich Publishers, New York Goult, R.J., R.F. Hoskins, Milner and M.J. Pratt, 1973.
Applicable mathematics for Engineers and Scientists. The Macmillan Press Ltd., London.
Ixaru, G.R. Liviu, 1984. Numerical methods for differential equations and applications. Editura Academiei, Bucuresti, Romania.
Chapman and Hall, New York.
Jain, R.K., 1984. Numerical solution of differential equations (2nd Edn.). Wiley Eastern Limited, New Delhi.
Jaun, A., 2001. Numerical methods for partial differential equations http://pde.fusion.kth.se
Kayode, S.J., 2004. A class of maximal order linear multistep collocation methods for direct solution of second order initial value problems of ordinary differential equations Ph.D Thesis, FUTA. (Unpublished).
Kayode, S.J., 2006. A P- stable collocation algorithm for direct solution of second order differential equations. To appear in International Journal of Numerical Mathematics.
